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Full text of "Men Of Mathematics"

MEN OF MATHEMATICS rnaticians plenty to do in developing their subject freely and somewhat uncritically — the Euclidean method was for long neglected in everything but geometry. We have already seen that the British school applied the method to algebra in the first half of the nineteenth century. Their successes seem to have made no very great impression on the work of then* contem- poraries and immediate successors, and it was only with the work of Hilbert that the postulational method came to be recognized as the clearest and most rigorous approach to any mathematical discipline. To-day this tendency to abstraction, in which the symbols and rules of operation in a particular subject are emptied of all meaning and discussed from a purely formal point of view, is all the rage, rather to the neglect of applications (practical or mathematical) which some say are the ultimate human justifi- cation for any scientific activity. Nevertheless the abstract method does give insights which looser attacks do not, and in particular the true simplicity of Boole's algebra of logic is most easily seen thus. Accordingly we shall state the postulates for Boolean algebra (the algebra of logic) and, having done so, see that they can indeed be given an interpretation consistent with classical logic. The following set of postulates is taken from a paper by E. V. Huntington, in the Transactions of the American Mathematical Society (vol. 85, 1933, pp. 274-304). The whole paper is easily understandable by anyone who has had a week of algebra, and may be found in most large public libraries. As Huntington points out, this first set of his which we transcribe is not as elegant as some of his others. But as its interpretation in terms of class inclusion as in formal logic is more immediate than the like for the others, it is to be preferred here. The set of postulates is expressed hi terms of -EC, +, X 9 where K is a class of undefined (wholly arbitrary, without any assigned meaning or properties beyond those given in the postu- lates) elements a, 6, c, ... , and a -f b and a x b (written also simply as ab) are the results of two undefined binary operations, -f, x ('binary', because each of-f, x operates on two elements of £). There are ten postulates, I a-VI: 490